On the crack inverse problem for pressure waves in half-space

Abstract

After formulating the pressure wave equation in half-space minus a crack with a zero Neumann condition on the top plane, we introduce a related inverse problem. That inverse problem consists of identifying the crack and the unknown forcing term on that crack from overdetermined boundary data on a relatively open set of the top plane. This inverse problem is not uniquely solvable unless some additional assumption is made. However, we show that we can differentiate two cracks Γ1 and Γ2 under the assumption that R3∖Γ1∪Γ2‾ is connected. If that is not the case we provide counterexamples that demonstrate non-uniqueness, even if Γ1 and Γ2 are smooth and “almost” flat. Finally, we show in the case where R3∖Γ1∪Γ2‾ is not necessarily connected that after excluding a discrete set of frequencies, Γ1 and Γ2 can again be differentiated from overdetermined boundary data.